Finite Group Databasebeta

← #149819

Subgroups — #149819

Order 512.

262 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20826C211yesmaximal, characteristic
2#11521C212yesmaximal
3#7373C212yesmaximal, complemented
4#7368C212yesmaximal, complemented
5#2920C411yescharacteristic
6#2920C2211yescharacteristic
7C22×C4×C8C411yescharacteristic
8#2165—22no—
9#1848C411yescharacteristic
10#1848C2211yescharacteristic
11#1837C411yescharacteristic
12#1837C2211yescharacteristic
13#1810—22no—
14C4×#108—22no—
15#1327C2214yes—
16#1320—22no—
17C23×C8—21no—
18C22×C42C2×C411yescharacteristic
19#432—41no—
20#432—41no—
21#431—41no—
22#431—41no—
23C2×#108—22no—
24C2×#108—22no—
25#423C2×C411yescharacteristic
26#423C2×C411yescharacteristic
27#423C2×C411yescharacteristic
28#423C2311yescharacteristic
29#407—21no—
30#407D412yes—
31C4×M16—21no—
32C4×M16D412yes—
33C2×C4×C8—21no—
34C2×C4×C8—22no—
35C2×C4×C8—22no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38#364—42no—
39#349—24no—
40#347—42no—
41#341—22no—
42#341—22no—
43C23×C4—21no—
44C23×C4C22⋊C411yescharacteristic
45C2×M16—21no—
46C2×M16—21no—
47C2×M16C22⋊C412yes—
48C2×M16C2×D412yes—
49C22×C8—21no—
50C22×C8—24no—
51C22×C8—21no—
52C22×C8—42no—
53C22×C8—42no—
54C2×C42—21no—
55C2×C42—21no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42C22⋊C411yescharacteristic
60C2×C42C22×C411yescharacteristic
61M32—44no—
62#108—44no—
63#108—44no—
64C4⋊C8—41no—
65C4⋊C8—41no—
66C4⋊C8—41no—
67C4⋊C8—41no—
68C22⋊C8—42no—
69C22⋊C8—42no—
70#97—22no—
71#97—41no—
72C4×C8—24no—
73C4×C8—22no—
74C4×C8—21no—
75C4×C8—21no—
76C4×C8—41no—
77C4×C8M1612yes—
78C4×C8M1612yes—
79C24#10111yescharacteristic
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—44no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4#10111yescharacteristic
92C22×C4C2×C22⋊C411yescharacteristic
93M16—42no—
94M16—42no—
95M16—42no—
96M16—42no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—48no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—41no—
108C42—21no—
109C42—21no—
110C42—21no—
111C42—21no—
112C42—21no—
113C42—21no—
114C42#9911yescharacteristic
115C42#9911yescharacteristic
116C42C2×M1611yescharacteristic
117C42C2×M1611yescharacteristic
118C16—84no—
119C23—21no—
120C23—42no—
121C23—41no—
122C23C2×#10111yescharacteristic
123C2×C4—24no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4#40711yescharacteristic
141C2×C4C2×#9911yescharacteristic
142C8—24no—
143C8—42no—
144C8—42no—
145C8—42no—
146C8—81no—
147C8—81no—
148C22—22no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22#146311yescharacteristic
156C4—22no—
157C4—22no—
158C4—21no—
159C4—41no—
160C4—41no—
161C4—41no—
162C4#212611yescharacteristic
163C4#213211yescharacteristic
164C2—21no—
165C2—42no—
166C2—41no—
167C2#1151311yescharacteristic
subgroupG/H
centerC4#2126
commutatorC2×C42C22×C4
Frattini, cosocle#423C23
socleC2#11513
prime radicalC24#101
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ #2920 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7373C2semidirect
4. #7368C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#208262
#115212
#73732
#73682